Methods Summary

In real-time, dividing deaths-to-date by cases-to-date leads to a biased estimate of the case fatality ratio (CFR), because this calculation does not account for delays from confirmation of a case to death, and under-reporting of cases.
Using the distribution of the delay from hospitalisation-to-death for cases that are fatal, we can estimate how many cases so far are expected to have known outcomes (i.e. death or recovery), and hence adjust the naive estimates of CFR to account for these delays.
The adjusted CFR does not account for under-reporting. However, the best available estimates of CFR (adjusting or controlling for under-reporting) are in the 1% - 1.5% range [1–4]. Large studies in China and South Korea estimating the CFR at 1.38% (95% CrI: 1.23–1.53%)[2] and 1.4% (95% CrI: 1.2-1.7%)[4] respectively. Based on these studies, and for simplicity, we assume a baseline CFR of 1.4% for our analysis.
If a country has an adjusted CFR that is higher (e.g. 20%), it suggests that only a fraction of cases have been reported (in this case, 1.420=7.0% cases reported approximately).

Current estimates for percentage of symptomatic cases reported for countries with greater than ten deaths

Temporal variation

Figure 1: Temporal variation in reporting rate. We calculate the percentage of symptomatic cases reported on each day a country has had more than ten deaths. We then fit a Gaussian Process (GP) to these data (see Temporal variation model fitting section for details), highlighting the temporal trend of each countries reporting rate. The red shaded region is the 95% CrI of fitted GP.

Adjusted symptomatic case estimates

Figure 2: Estimated number of new symptomatic cases, calculated using our temporal under-reporting estimates. We adjust the reported case numbers each day - for each country with an under-reporting estimate - using our temporal under-reporting estimates to arrive at an estimate of the true number of symptomatic cases each day. The shaded blue region represents the 95% CrI, calcuated directly using the 95% CrI of the temporal under-reporting estimate.

Reported cases

Figure 3: Reported number of cases each day, pulled from the ECDC and plotted against time for comparison with our estimated true numbers of symptomatic cases each day, adjusted using our under-reporting estimates.

Table of current estimates

Country	Percentage of symptomatic cases reported (95% CI)	Total cases	Total deaths
Afghanistan	44% (31%-60%)	7,072	173
Albania	81% (42%-100%)	948	31
Algeria	33% (21%-48%)	7,201	555
Andorra	22% (12%-37%)	761	51
Argentina	20% (15%-24%)	8,358	382
Armenia	78% (54%-99%)	4,823	61
Australia	86% (53%-100%)	7,060	99
Austria	45% (28%-70%)	16,179	629
Azerbaijan	85% (59%-100%)	3,287	40
Bahamas	50% (12%-98%)	96	11
Bangladesh	82% (44%-100%)	23,870	349
Belarus	99% (93%-100%)	30,572	171
Belgium	12% (8.8%-15%)	55,559	9,080
Bolivia	19% (14%-25%)	4,263	174
Bosnia and Herzegovina	12% (6.8%-19%)	2,303	132
Brazil	13% (10%-15%)	254,220	16,792
Bulgaria	27% (18%-41%)	2,259	112
Burkina Faso	25% (15%-46%)	796	51
Cameroon	33% (16%-58%)	3,292	140
Canada	14% (11%-16%)	78,061	5,842
Chad	7.3% (4.1%-13%)	519	53
Chile	93% (79%-100%)	46,059	478
China	98% (70%-100%)	84,063	4,638
Colombia	29% (23%-37%)	16,295	592
Congo	30% (9.7%-74%)	412	15
Cote dIvoire	83% (53%-100%)	2,119	28
Croatia	19% (10%-31%)	2,228	95
Cuba	35% (22%-57%)	1,881	79
Cyprus	83% (48%-100%)	917	17
Czechia	33% (24%-43%)	8,586	297
Democratic Republic of the Congo	34% (19%-57%)	1,629	61
Denmark	34% (24%-46%)	10,968	548
Dominican Republic	50% (37%-65%)	12,725	434
Ecuador	8.9% (6.7%-11%)	33,582	2,799
Egypt	34% (26%-43%)	12,764	645
El Salvador	49% (27%-86%)	1,413	30
Estonia	33% (21%-51%)	1,784	64
Finland	32% (19%-50%)	6,380	300
France	7.9% (6.4%-9.4%)	142,903	28,239
Germany	21% (17%-26%)	175,210	8,007
Ghana	96% (83%-100%)	5,735	29
Greece	29% (18%-44%)	2,836	165
Guatemala	46% (28%-76%)	2,001	38
Guernsey	43% (13%-94%)	252	13
Guinea	90% (65%-100%)	2,796	16
Guyana	53% (11%-99%)	124	10
Haiti	8.6% (4.4%-16%)	533	21
Honduras	20% (13%-33%)	2,798	146
Hungary	12% (8.1%-16%)	3,556	467
Iceland	87% (55%-100%)	1,802	10
India	37% (30%-45%)	101,139	3,163
Indonesia	27% (17%-38%)	18,010	1,191
Iran	36% (29%-43%)	122,492	7,057
Iraq	57% (29%-93%)	3,554	127
Ireland	42% (29%-57%)	24,200	1,547
Isle of Man	23% (8.2%-72%)	335	24
Israel	81% (62%-99%)	16,621	272
Italy	14% (11%-16%)	225,886	32,007
Japan	14% (10%-19%)	16,365	763
Jersey	13% (6.2%-29%)	303	27
Kazakhstan	97% (86%-100%)	6,751	35
Kenya	16% (9.6%-25%)	912	50
Kosovo	49% (28%-83%)	955	29
Kuwait	85% (64%-100%)	15,691	118
Kyrgyzstan	69% (34%-100%)	1,243	14
Latvia	50% (24%-93%)	1,009	19
Lebanon	58% (30%-98%)	931	26
Liberia	47% (11%-97%)	229	22
Lithuania	34% (19%-53%)	1,547	59
Luxembourg	51% (33%-71%)	3,947	107
Malaysia	95% (71%-100%)	6,941	113
Mali	20% (13%-31%)	874	52
Mauritius	62% (16%-100%)	332	10
Mexico	7.2% (5.8%-8.5%)	51,633	5,332
Moldova	29% (21%-38%)	6,138	215
Morocco	96% (82%-100%)	6,952	192
Netherlands	16% (13%-20%)	44,141	5,694
New Zealand	47% (23%-86%)	1,153	21
Niger	16% (7.2%-29%)	909	55
Nigeria	32% (22%-44%)	6,175	191
North Macedonia	26% (17%-41%)	1,817	104
Norway	38% (17%-66%)	8,249	233
Oman	95% (81%-100%)	5,379	26
Pakistan	48% (37%-59%)	43,966	939
Panama	51% (37%-68%)	9,726	279
Paraguay	84% (51%-100%)	788	11
Peru	37% (30%-45%)	94,933	2,789
Philippines	19% (14%-25%)	12,718	831
Poland	25% (19%-31%)	18,885	936
Portugal	41% (32%-51%)	29,209	1,231
Puerto Rico	36% (23%-55%)	2,710	124
Qatar	91% (59%-100%)	33,969	15
Romania	18% (14%-23%)	17,036	1,107
Russia	91% (80%-99%)	290,678	2,722
San Marino	78% (41%-100%)	654	41
Saudi Arabia	99% (93%-100%)	57,345	320
Senegal	72% (42%-99%)	2,544	26
Serbia	88% (62%-100%)	10,699	231
Sierra Leone	11% (6.3%-19%)	519	33
Singapore	94% (70%-100%)	28,343	22
Sint Maarten	12% (4%-33%)	77	15
Slovakia	66% (38%-98%)	1,495	28
Slovenia	17% (11%-27%)	1,466	104
Somalia	35% (19%-61%)	1,455	57
South Africa	42% (33%-54%)	16,433	286
South Korea	47% (20%-83%)	11,078	263
Spain	14% (11%-17%)	231,606	27,709
Sudan	20% (13%-30%)	2,591	105
Sweden	10% (8.1%-12%)	30,377	3,698
Switzerland	23% (18%-29%)	30,514	1,602
Thailand	75% (50%-99%)	3,031	56
Tunisia	57% (27%-98%)	1,043	46
Turkey	65% (53%-79%)	150,593	4,171
Ukraine	36% (23%-49%)	18,616	535
United Arab Emirates	92% (75%-100%)	24,190	224
United Kingdom	16% (13%-19%)	246,406	34,796
United Republic of Tanzania	46% (24%-83%)	509	21
United States of America	20% (16%-24%)	1,508,598	90,353
Uruguay	45% (21%-86%)	737	20
Uzbekistan	91% (67%-100%)	2,802	13
Venezuela	82% (45%-100%)	618	10

Table 1: Estimates for the proportion of symptomatic cases reported in different countries using cCFR estimates based on case and death timeseries data from the ECDC. Total cases and deaths in each country is also shown. Confidence intervals calculated using an exact binomial test with 95% significance.

Adjusting for outcome delay in CFR estimates

During an outbreak, the naive CFR (nCFR), i.e. the ratio of reported deaths date to reported cases to date, will underestimate the true CFR because the outcome (recovery or death) is not known for all cases [5]. We can therefore estimate the true denominator for the CFR (i.e. the number of cases with known outcomes) by accounting for the delay from confirmation-to-death [1].

We assumed the delay from confirmation-to-death followed the same distribution as estimated hospitalisation-to-death, based on data from the COVID-19 outbreak in Wuhan, China, between the 17th December 2019 and the 22th January 2020, accounting right-censoring in the data as a result of as-yet-unknown disease outcomes (Figure 1, panels A and B in [7]). The distribution used is a Lognormal fit, has a mean delay of 13 days and a standard deviation of 12.7 days [7].

To correct the CFR, we use the case and death incidence data to estimate the proportion of cases with known outcomes [1,6]:

\[ u_{t} = \frac{ \sum_{j = 0}^{t} c_{t-j} f_j}{c_t}, \]

where \(u_t\) represents the underestimation of the proportion of cases with known outcomes [1,5,6] and is used to scale the value of the cumulative number of cases in the denominator in the calculation of the cCFR, \(c_{t}\) is the daily case incidence at time, \(t\) and \(f_t\) is the proportion of cases with delay of \(t\) between confirmation and death.

Approximating the proportion of symptomatic cases reported

At this stage, raw estimates of the CFR of COVID-19 correcting for delay to outcome, but not under-reporting, have been calculated. These estimates range between 1% and 1.5% [1–3]. We assume a CFR of 1.4% (95% CrI: 1.2-1.7%), taken from a recent large study [3], as a baseline CFR. We use it to approximate the potential level of under-reporting in each country. Specifically, we perform the calculation \(\frac{1.4\%}{\text{cCFR}}\) of each country to estimate an approximate fraction of cases reported.

Temporal variation model fitting

We estimate the level of under-reporting on every day for each country that has had more than ten deaths. We then fit a Gaussian Process (GP) model using the library greta and greta.gp. The parameters we fit and their priors are the following: \[ \begin{aligned} &\sigma \sim \text{Log Normal(-1, 1)}: \quad &\text{Variance of the reporting kernel} \\ &\text{L} \sim \text{Log Normal(4, 0.5)}: \quad &\text{Lengthscale of the reporting kernel} \\ &\sigma_{\text{obs}} \sim \text{Truncated Normal(0, 0.5)}, \quad &\text{Variance of the obseration kernel, truncated at 0} \end{aligned} \] The kernel is split into two components: the reporting kernel \(R\), and the observation kernel \(O\). The reporting component has a standard squared-exponential form. For the observation component, we use an i.i.d. noise kernel to acccount for observation overdispersion, which can smooth out overly clumped death time-series. This is important as some countries have been known to report an unusually large number of deaths on a single day, due to past under-reporting.

In the sampling and fitting process, we calculate the expected number of deaths at each time-point, given the baseline CFR. We then use a Poisson likelihood, where the expected number of deaths is the rate of the Poisson likelihood, given the observed number of deaths

Adjusting case counts for under-reporting

We adjust the reported number of cases each day, pulled from the ECDC. Specifically, we divide the case numbers of each day by our “proportion of cases reported” estimates that we calculate each day for each country.*

Limitations

Implicit in assuming that the under-reporting is \(\frac{1.4\%}{\text{cCFR}}\) for a given country is that the deviation away from the assumed 1.4% CFR is entirely down to under-reporting. In reality, burden on healthcare system is a likely contributing factor to higher than 1.4% CFR estimates, along with many other country specific factors.

The following is a list of the other prominent assumptions made in our analysis:

We assume that people get tested upon hospitalisation. A few examples where this is not the case are Germany and South Korea, where people can get tested earlier.
We assume that hospitalisation to death from early Wuhan is representative of the all the other countries (by using the distribution parameterised using early Wuhan data) and that all countries have the same risk and age profile as Wuhan.
Severity of COVID-19 is known to increase with age. Therefore, countries with older populations will naturally see higher death rates. We are extending this analysis to adjust for the age distribution for countries with more than five reported deaths and where age distribution data is available.
All results are linked and biased by the baseline CFR, assumed at 1.4% [3].
The under-reporting estimate is sensitive to the baseline CFR, meaning that small errors in it lead to large errors in the estimate for under-reporting.
There are several sources of uncertainty in this analysis: the reported mean and SD of the delay distribution; the cCFR for each country (on each day) and the baseline CFR. We use the lower and upper mean and SD of the delay distribution to produce the widest interval. The other two uncertainties are just carried through directly in the subsequent calculations, which is crude. An ongoing extension is developing this into a fully Bayesian model, which deals with these different uncertainties more rigorously.

Code and data availability

The code is publically available at https://github.com/thimotei/CFR_calculation. The data required for this analysis is a time-series for both cases and deaths, along with the corresponding delay distribution. We scrape this data from ECDC, using the NCoVUtils package [8].

Acknowledgements

The authors, on behalf of the Centre for the Mathematical Modelling of Infectious Diseases (CMMID) COVID-19 working group, wish to thank DSTL for providing the High Performance Computing facilities and associated expertise that has enabled these models to be prepared, run and processed and in an appropriately-rapid and highly efficient manner.

References

1 Russell TW, Hellewell J, Jarvis CI et al. Estimating the infection and case fatality ratio for covid-19 using age-adjusted data from the outbreak on the diamond princess cruise ship. medRxiv 2020.

2 Verity R, Okell LC, Dorigatti I et al. Estimates of the severity of covid-19 disease. medRxiv 2020.

3 Guan W-j, Ni Z-y, Hu Y et al. Clinical characteristics of coronavirus disease 2019 in china. New England Journal of Medicine 2020.

4 Shim E, Mizumoto K, Choi W et al. Estimating the risk of covid-19 death during the course of the outbreak in korea, february-march, 2020. medRxiv 2020.

5 Kucharski AJ, Edmunds WJ. Case fatality rate for ebola virus disease in west africa. The Lancet 2014;384:1260.

6 Nishiura H, Klinkenberg D, Roberts M et al. Early epidemiological assessment of the virulence of emerging infectious diseases: A case study of an influenza pandemic. PLoS One 2009;4.

7 Linton NM, Kobayashi T, Yang Y et al. Incubation period and other epidemiological characteristics of 2019 novel coronavirus infections with right truncation: A statistical analysis of publicly available case data. Journal of Clinical Medicine 2020;9:538.

8 Abbott S MJ Hellewell J. NCoVUtils: Utility functions for the 2019-ncov outbreak. doi:105281/zenodo3635417 2020.

Aim